Questions regarding early natural numbers.

Question

Consider the real number $0.123456789101112\dots$, where you concatenate the digits of the natural numbers.

Certain natural numbers are "early", meaning, they appear earlier as a substring of digits than they are supposed to.

For example: $12$, $23$ and $101$ are early natural numbers.

I have a few questions:

1. Is the set of early numbers infinite? What about the set of non-early numbers?

2. Does the set of early numbers have an asymptotic density? If so, is there a closed form expression for that real number?

3. Same as previous question, but regarding the set of non-early numbers.

Answer 1

These numbers are tabulated at the Online Encyclopedia of Integer Sequences. There's a reference there to work of Golomb, which you can find on page 30 of this link. In particular, Golomb explains why these numbers have density 1.

Answer 2

The set of early numbers is infinite, as it contains at least the following set: $$\{12,123,1234,12345,123456,1234567,\dots\}$$

As for the set of non-early numbers, I think you can show that the number $1111\dots 1$ is not early for any number of ones it contains, meaning the set of non-early numbers is also infinite.

Questions 2 and 3 are much tougher and I cannot see a simple answer to them (still, I believe I answered at least some of your questions).

Answer 3

The set of early numbers will definitely go infinite because you can group certain number of digits to achieve a early number. There are also non-early numbers which will go to infinite. There would be always certain numbers (more than early ones) who can't be achieved by the combination of any numbers.